On the controllability of the Navier-Stokes equation in a rectangle, with a little help of a distributed phantom force

On the con trollabilit y of the Na vier-Stok es equation in a rectangle, with a little help of a distributed phan tom force Jean-Michel Cor on Fr ´ ed ´ eric Marba ch Franck Sueur Ping Zhang Abstract. This note ec ho es the talk giv en by the second author during the Journ ´ ees EDP 2018 in Ob ernai. Its aim is to provide an o verview and a sketc h of pro of of the result obtained b y the authors in [6], concerning the controllabilit y of the Navier-Stok es equation. W e refer the in terested readers to the original paper for the full technical details of the proof, which will b e omitted here, to fo cus on the main underlying ideas. 1. Geometric setting W e consider a rectangular domain Ω := (0 , L ) × ( − 1 , 1), where L > 0 is the horizon tal length of the domain (see Figure 1). W e will use ( x, y ) ∈ Ω as co ordinates. W e see this rectangular domain as a tub e or a river, in the interior of which a fluid evolv es. During some time in terv al [0 , T ], the ev olution of the fluid velocity u ( t, x, y ) is gov erned b y the homogeneous incompressible Na vier-Stokes equation:  ∂ t u + ( u · ∇ ) u − ∆ u + ∇ p = f , div u = 0 , (1.1) where f ( t, x, y ) is a small external vectorial forcing term, whose role will b e explained below and p ( t, x, y ) is the scalar pressure field corresp onding to the incompressibility constraint. x y 0 Γ − Γ + L Γ 0 Γ L Ω Figure 1. Ph ysical domain Ω On the upp er and lo w er horizon tal boundaries Γ ± := (0 , L ) × {± 1 } , corresp onding to the w alls of the tub e or the banks of the riv er, w e assume that the fluid satisfies the usual Keywor ds: Navier-Stok es, Controllabilit y . Math. classific ation: 35Q30, 93B05, 93C20. 1 J.-M. Cor on, F. Marba ch, F. Sueur, & P. Zhang no-slip Dirichlet b oundary condition: u = 0 . (1.2) Con versely , a key feature of the geometric setting at stake is that no b oundary condition is prescrib ed a priori on the left and right v ertical b oundaries Γ 0 := { 0 } × ( − 1 , 1) and Γ L := { L } × ( − 1 , 1). This under-determination mo dels the idea that w e can act on the system by exerting some forcing (sa y , through suction or blowing actions) on the fluid. 2. Cauc h y problem In 2D, it is known that weak Leray solutions to the homogeneous incompressible Na vier- Stok es equation exist globally and are unique. In our setting, uniqueness is not guaran teed b ecause the problem is under-determined due to the p ossible choi ces of the b oundary conditions on Γ 0 and Γ L (whic h corresp ond to controls). More precisely , let L 2 div (Ω) denote the space of L 2 v ector fields on Ω which are divergence- free and tangen t to the b oundaries Γ ± . Giv en T > 0, an initial data u ∗ ∈ L 2 div (Ω), and a forcing f ∈ L 1 ((0 , T ); L 2 (Ω)), we will sa y that u ∈ C 0 ([0 , T ]; L 2 div (Ω)) ∩ L 2 ((0 , T ); H 1 (Ω)) is a w eak Leray solution to (1.1) and (1.2) with final data u T ∈ L 2 div (Ω) when it satisfies the weak formulation: − Z T 0 Z Ω u · ∂ t ϕ + Z T 0 Z Ω ( u · ∇ ) u · ϕ + 2 Z T 0 Z Ω D ( u ) : D ( ϕ ) = Z Ω u ∗ · ϕ (0 , · ) − Z Ω u T · ϕ ( T , · ) + Z T 0 Z Ω ϕ · f , (2.1) for every test function ϕ ∈ C ∞ ([0 , T ] × ¯ Ω) which is divergence-free, tangen t to Γ ± and v anishes on Γ 0 and Γ L . Another w ay to form ulate the Cauch y problem is to see weak Leray solutions on Ω as the restriction to the ph ysical domain Ω of weak solutions defined on a larger domain, sa y the strip B := R × ( − 1 , 1), corresp onding to some extensions of the initial data and of the external force. Giv en any (reasonable) choice of extensions for u ∗ and f , there exists a unique global weak solution on B , whic h can then b e restricted to Ω. 3. A conjecture of Lions In the late 1980’s, Jacques-Louis Lions formulated m ultiple open problems and conjectures concerning the con trollability of systems gov erned by partial differen tial equations. In particular, in [13], he asked whether the Navier-Stok es equation was small-time globally n ull con trollable. There are many w ays to set this question, dep ending on the geometry , on the exact goals, and on the nature of the exerted controls (which can either b e a distributed force in some strict subset of the domain or come in to play through b oundary data). In our geometrical setting, the conjecture of con trollability can b e form ulated as: Conjecture 3.1. L et T > 0 and u ∗ ∈ L 2 div (Ω) . Ther e exists a we ak L er ay solution to (1.1) with f = 0 and (1.2) such that the final state satisfies u ( T , · ) = 0 . The difficult y in the question comes from the combination of m ultiple factors. First, the allotted con trol time T > 0 may b e v ery small, whic h requires to use an asymptotically rapid strategy . Second, the initial data u ∗ ma y b e very large, so that the nonlinearit y in the Navier-Stok es equation plays an imp ortan t role. Last, but not least, the con trols are only exerted on a strict subset Γ 0 ∪ Γ L of the full b oundary ∂ Ω. One can exp ect that sp ecific phenomenons o ccur near the uncon trolled parts Γ ± . 2 NA VIER-STOKES CONTROLLABILITY 4. Our main controllabilit y result In [6], w e pro ved a result whic h almost brings a p ositiv e answ er to the abov e conjecture. Whereas the initial conjecture implies to find an exact solution of the Na vier-Stok es equa- tion with a null forcing term, we in tro duce a non-zero but arbitrarily small forcing, in arbitrarily strong norms. Theorem 4.1. L et T > 0 and u ∗ ∈ L 2 div (Ω) . F or every k ∈ N and every η > 0 , ther e exists a for c e f ∈ L 1 ((0 , T ); H k (Ω)) satisfying k f k L 1 ((0 ,T ); H k (Ω)) ≤ η , (4.1) and an asso ciate d we ak L er ay solution u ∈ C 0 ([0 , T ]; L 2 div (Ω)) ∩ L 2 ((0 , T ); H 1 (Ω)) to (1.1) and (1.2) satisfying u (0) = u ∗ and u ( T ) = 0 . In this under-determined formulation of the control result, the b oundary con trols (i.e. the traces of u on the b oundaries Γ 0 and Γ L ) are not explicitly written. Hence, we almost obtain small-time global exact n ull con trollability . Our metho d does not easily extend to obtain the “true” control result with f = 0. Indeed, one cannot pass to the limit in the main theorem b ecause there is no a priori b ound on the size of the tra jectories u as η → 0. The small correction we need is link ed with our pro of strategy (which creates a b oundary la yer) and our pro of technique (which relies on horizontal analyticity). It is likely that pro ving the result for f = 0 requires b oth a new strategy and a new technique. The fact that Ω = (0 , L ) × ( − 1 , 1) is a “flat” domain is also very imp ortant for our pro of. More precisely , the k ey p oin t is that the uncontrolled b oundaries Γ ± are flat in the horizon tal direction. This feature allows us to introduce almost explicit expressions for some of the profiles that build up the solution u , whic h are solutions to linear equations. 5. Discussion on earlier results The open problems introduced b y Jacques-Louis Lions concerning con trollabilit y for fluid mec hanics problems hav e received a large attention. Small initial data and local results. Small-time lo cal n ull controllabilit y w as already kno wn. F or ev ery T > 0, there exists δ T > 0 such that, for every u ∗ ∈ L 2 (Ω) satisfying k u ∗ k ≤ δ T , one can find controls driving u ∗ to the null equilibrium state u = 0 in time T . This can be done using only b oundary controls, without any distributed force ( f = 0). In this case, since the state is small, one sees the bilinear term in the Navier-Stok es system as a small p erturbation term of the Stok es equation so that the con trollability is pro ved thanks to Carleman estimates and fixed p oin t theorems. Lo osely sp eaking, such an approac h corresp onds to low Reynolds controllabilit y . W e refer to [7, 8, 12] for some imp ortan t contributions to this topic, successively impro ving the smallness assumptions, the control domains or the reac hable targets. Global results without b oundaries. F or large initial data, a setting corresp onding to con trollability at large Reynolds n umbers, the first author and F ursiko v pro v ed global null con trollability for the Na vier-Stokes system in a 2D manifold without b oundary in [3] (in this case, the con trol is an internal con trol exerted from a small open subset of the domain). In [9], F ursik ov and Imanuvilo v pro v ed a small-time global control result when the control is exerted on the full b oundary ∂ Ω of the physical fluid domain. Both geometries share the imp ortan t feature that there is no uncontrolled p ortion of the b oundary . 3 J.-M. Cor on, F. Marba ch, F. Sueur, & P. Zhang Na vier slip-with-friction b oundary condition. Jacques-Louis Lions’ problem has b een solved in [4] by the first three authors in the particular case of the Navier slip- with-friction b oundary condition (see also [5] for a gen tle in tro duction to this result). This b oundary condition is less stringen t than (1.2) since it allo ws the fluid to slide tangen tially along the b oundary . In this context, small-time global exact n ull con trollability and small- time global exact con trollabilit y to tra jectories hold for ev ery regular domain (2D and 3D) and for ev ery subset of the domain where the con trol is exerted, pro vided that it intersects eac h connected comp onen t of the b oundary of the ph ysical domain. P artial results with large forcing. The closest works to Theorem 4.1 are refer- ences [10, 11], in which related results are obtained in v ery similar settings. These w orks pro ve a v ersion of Theorem 4.1 for which the distributed force f can be c hosen small in L p ((0 , T ); H − 1 (Ω)), where 1 < p < 4 / 3. The fact that our phantom force can be chosen arbitrarily small in the space L 1 ((0 , T ) , H k (Ω)) for an y k ≥ 0, is the ma jor impro vemen t of this work. In particular, b eing small, say in C 1 ( ¯ Ω) guarantees that there is not fast scale v ariations of our distributed force near the uncon trolled boundaries. This p ossibilit y is not ruled out b y a conclusion on the smallness in H − 1 (Ω) of the forcing term. 6. A strategy based on the flushing of the v ortexes If one thinks that the v ector field u ( t, · ) is describ ed by the com bination of its potential part and its v orticity , driving to zero requires to driv e b oth parts to zero. Thanks to the incompressibility constrain t, it is v ery easy to make the p oten tial part v anish, almost instan tly . Indeed, if one chooses n ull boundary con trols on Γ 0 and Γ L , then at an y instan t t > 0, the full state u ( t, · ) can be recov ered from its vorticit y ω ( t, · ) through the follo wing div-curl problem:      curl u ( t, · ) = ω ( t, · ) in Ω , div u ( t, · ) = 0 in Ω , u ( t, · ) · n = 0 on ∂ Ω . (6.1) W e can th us assume that the initial data has a v anishing av erage horizon tal v elo cit y , i.e. R Ω u ∗ · e x = 0, where e x is the tangential unit v ector. If it is not the case, using such n ull con trols will ensure it for an y p ositiv e time. W e em b ed Ω in the band B = R × ( − 1 , +1) and extend the initial data u ∗ to a compactly supp orted (say on [ − L, 2 L ] × [ − 1 , 1]) div ergence-free initial data on B (this is p ossible when u ∗ has zero a verage tangen tial sp eed), which we will still denote b y u ∗ . W e work in the extended domain B for simplicity . Our goal is th us to build a solution suc h that u ( T ) | Ω = 0. In fact, it is sufficient to ac hieve k u ( T ) | Ω k L 2 (Ω)  1, since lo cal controllabilit y is known for the Na vier-Stokes equation (see the paragraph Smal l initial data and lo c al r esults of Section 5). Recalling that, in 2D, the vorticit y is transp orted b y the flo w, the first imp ortan t idea is to flush the support of the initial v orticity ω ∗ := curl u ∗ outside of the initial ph ysical domain Ω, into the extension B \ Ω. W e p erform this task using the incompressibilit y and in tro ducing artificially a high pressure gradient as sk etched in Figure 2. 7. Asymptotic implemen tation of the flushing metho d In order to implemen t the intuition of Figure 2, we in tro duce a small parameter ε > 0 and w e will construct a solution u ( t, x, y ) giv en under the form u ( t, x, y ) = 1 ε u ε  t ε , x, y  , (7.1) 4 NA VIER-STOKES CONTROLLABILITY curl u ∗ curl u ∗ Ω curl u = 0 t = 0 t = T Figure 2. Flushing pro cess for the vorticit y where the new unknown u ε m ust now solv e the following mo dified equation on a larger time interv al t ∈ (0 , T /ε )      ∂ t u ε + ( u ε · ∇ ) u ε − ε ∆ u ε + ∇ p ε = f ε , div u ε = 0 , u ε (0) = εu ∗ , (7.2) where w e introduced p ε ( t, x, y ) = ε 2 p ( εt, x, y ) and f ε ( t, x, y ) = ε 2 f ( εt, x, y ). Within this scaling, the goal is to construct a solution suc h that k u ε ( T /ε ) | Ω k L 2 (Ω)  ε . Heuristically , w e wish to build a solution to (7.2) which b eha v es as u ε ( t, x, y ) ≈ h ( t ) e x + εu 1 ( t, x, y ) + o ( ε ) , (7.3) where h ∈ C ∞ ( R + ; R ) is supp orted on (0 , T ) and has a sufficiently large in tegral, sa y R T 0 h ( t )d t ≥ 3 L , and u 1 is the solution to the linearized version of (7.2) around h ( t ) e x (whic h is a solution of the underlying Euler equation, see the red arrows on Figure 2),      ∂ t u 1 + h∂ x u 1 + ∇ p 1 = 0 , div u 1 = 0 , u 1 (0) = u ∗ . (7.4) Of course, thanks to the simple geometrical setting, (7.4) can b e solved explicitly as u 1 ( t, x, y ) = u ∗  x − Z t 0 h ( t 0 )d t 0 , y  . (7.5) In particular, if u ∗ w as compactly supp orted, say on [ − L, 2 L ] × [ − 1 , 1] ⊂ ¯ B , then u 1 v anishes inside Ω for t ≥ T thanks to the assumption that R h ≥ 3 L . If we b eliev e that the remainder in (7.3) is indeed o ( ε ), then the theorem is prov ed since, for t ≥ T (th us including t = T /ε ), h ( t ) v anishes and u 1 ( t ) v anishes inside Ω, so that k u ε ( T /ε ) | Ω k L 2 (Ω)  ε . 8. T angen tial b oundary lay ers Unfortunately , the leading order profile h ( t ) e x is the solution of the underlying Euler equation (corresponding to ε = 0 in (7.2)) and only satisfies the normal imp ermeabilit y condition u · e y = 0 on Γ ± . The tangential b oundary condition u · e x = 0 is not satisfied b y this profile. Hence, there is no c hance for an expansion like (7.3) to hold. 5 J.-M. Cor on, F. Marba ch, F. Sueur, & P. Zhang This discrepancy is v ery usual when studying the con vergence of Navier-Stok es to Euler in the v anishing viscosit y setting. It gives rise to the theory of b oundary la yers: a small region, here of width ε 1 2 , within which the viscous effects remain important and allo wing to recov er the missing b oundary condition. Plugging suc h an Ansatz dep ending on a fast v ariable in the Na vier-Stokes equations yields the Prandtl equation [14] gov erning the ev olution of the b oundary la yer profile. Here, thanks to the flat geometric setting and the inv ariance with resp ect to x of the main profile, they tak e a particularly simple form. Indeed, we change our expansion (7.3) into u ε ( t, x, y ) ≈  h ( t ) − V  t, 1 + y √ ε  e x + εu 1 ( t, x, y ) + o ( ε ) , (8.1) where V : R + × R + → R is the solution to the following heat equation (a v ery simplified v ersion of the Prandtl equation in our setting):      ∂ t V + ∂ z z V = 0 , V ( t, 0) = h ( t ) , V (0 , z ) = 0 . (8.2) In fact, a second symmetrical corrector dep ending on 1 − y is required in order to accoun t for the b oundary la yer near Γ + , and smo oth slo wly v arying cutoff functions are needed in order to av oid interaction b et w een the t w o correctors. W e will not consider these details here and pro ceed with the computations only with the corrector near Γ − , as they already con tain the core ideas. These correctors allo w to build a reference flo w which fully satisfies the b oundary con- ditions on Γ ± , enabling us to hop e to pro v e (8.1). 9. Main difficulties Although the b oundary correctors only change the v alue of the reference flow in small strips near the b oundaries, they in tro duce tw o imp ortan t difficulties with resp ect to our con trollability goal. First, at the final time t = T /ε , although h ( t ) v anishes and u 1 ( t ) v anishes inside Ω, it is not the case for V . More precisely , one has   u ( T ) | Ω   L 2 (Ω) =      1 ε u ε  T ε  | Ω      L 2 (Ω) ≈ ε − 3 4     V  T ε      L 2 ( R + ) . (9.1) F or t ≥ T , the heat equation (8.2) has zero source term and the profile V deca ys in L 2 ( R + ). Unfortunately , without an y additional assumption, studying the decay rates for the free heat equation on the half line only yields a w eak deca y of the form k V ( t ) k L 2 ( R + ) ≤ C t − 1 4 as t → + ∞ , whic h is not sufficien t to counterbalance the prefactor of (9.1). Second, trying to make expansion (8.1) rigorous and computing the equation satisfied b y the remainder r ε (for u ε = h + V + εu 1 + εr ε ), yields an evolution equation with a bad amplification term:      ∂ t r ε + ε − 1 2 r ε 2 ∂ z V e x + A ε r ε + ε ( r ε · ∇ ) r ε − ε ∆ r ε = σ ε , div r ε = 0 , r ε (0) = 0 . (9.2) In (9.2), σ ε is a small source term in some appropriate sense (one can think σ ε = o (1) in L 1 ((0 , T /ε ); L 2 ( B )) for example). The amplification has a reasonable part A ε r ε (one can think that its norm in L 1 ((0 , T /ε ); L ∞ ( B )) is b ounded uniformly with resp ect to ε ) and a v ery bad part ε − 1 2 r ε 2 ∂ z V e x . Performing naiv e Gr¨ on wall estimates on this equation 6 NA VIER-STOKES CONTROLLABILITY is therefore b ound to fail due to this term, ev en more so since we intend to perform these estimates up to the large final time T /ε (w e w ould then exp ect an exp onen tial amplification of the form exp( ε − 3 / 2 )). So, a priori , the remainder is not small. 10. Recasting amplification as a loss of deriv ativ e W e start b y dealing with the second problem. Using the div ergence free condition on r ε and the n ull b oundary condition on r ε 2 , we wish to rewrite the amplification term. W e p erform the computation near the lo w er wall y = − 1. Near this w all, V is ev aluated at z = ε − 1 2 (1 + y ). Hence one has ε − 1 2 r ε 2 ( t, x, y ) ∂ z V ( t, z ) e x = ε − 1 2 (1 + y )  1 1 + y Z y − 1 ∂ y r ε 2 ( t, x, y 0 )d y 0  ∂ z V ( t, z ) e x = −  1 1 + y Z y − 1 ∂ x r ε 1 ( t, x, y 0 )d y 0  z ∂ z V ( t, z ) e x . (10.1) Th us, the amplification term has b een recast as a lo cal a verage in the normal direction of ∂ x r ε 1 . Ho wev er, this term do es not hav e the structure of a transp ort term that would disapp ear during energy estimates b y in tegration b y parts. On a formal lev el, one should rather think of this term as the structure-less term ( z ∂ z V ) | ∂ x | r ε , (10.2) where | ∂ x | is defined as the F ourier m ultiplier by | ξ | , where ξ is the horizontal F ourier v ariable. Since V is the solution to (8.2), for eac h t ≥ 0, the map z 7→ z ∂ z V ( t, z ) b elongs to L ∞ ( R + ) b ecause V ( t, · ) and its deriv atives deca y exponentially (with respect to z → + ∞ ). There is a priori no hop e to “absorb” a term such as (10.2) by the − ε ∆ r ε dissipation term of (9.2), b ecause the estimate would once again degenerate as ε → 0. Instead, we think of (10.2) as a loss of deriv ative, and w e will w ork in an analytic setting (with resp ect to the tangential v ariable), so that lo osing one deriv ative (among an infinite num b er of deriv atives) is not to o bad. In the con text of Na vier-Stokes b oundary la yers, analyticit y w as first used in [15, 16] to pro ve b oth the existence of solutions to the Prandtl equation and the conv ergence of the v anishing viscosit y Na vier-Stokes solution to an Euler+Prandtl system for analytic data. 11. Cauc h y-Kow alesk ay a schema Due to the term (10.2), the analytic radius of the solution r ε will deca y as time increases. This rough idea can be very precisely quan tified thanks to an idea link ed with Cauc hy- Ko walesk a ya t yp e theorems. Let ρ ∈ C 1 ( R + ; R ). W e introduce the new unkno wn r ε ρ := e ρ ( t ) | ∂ x | r ε . (11.1) This change of unkno wn is licit for example when the tangential F ourier transform of r ε is supp orted on some b ounded region − N ≤ ξ ≤ N , so this trick has to b e p erformed on a “frequency-truncated” version of (9.2), which will then pass to the limit since the resulting estimates will not dep end on N . Under the change of unkno wn (11.1), equation (9.2) is roughly changed into ∂ t r ε ρ − ρ 0 ( t ) | ∂ x | r ε ρ − ( z ∂ z V ) | ∂ x | r ε ρ = ... (11.2) Therefore, multiplying (11.2) b y r ε ρ and using P arsev al’s formula yields 1 2 d d t Z ( ˆ r ε ρ ) 2 − ρ 0 ( t ) Z | ξ | ( ˆ r ε ρ ) 2 ≤ k z ∂ z V ( t, z ) k L ∞ ( R + ) Z | ξ | ( ˆ r ε ρ ) 2 + ..., (11.3) 7 J.-M. Cor on, F. Marba ch, F. Sueur, & P. Zhang so that the deriv ativ e loss term of the righ t-hand side can b e absorb ed if and only if − ρ 0 ( t ) ≥ k z ∂ z V ( t, z ) k L ∞ ( R + ) . (11.4) Since (11.4) m ust b e satisfied for t ∈ [0 , T /ε ] and since w e wish ρ to sta y p ositiv e, w e need to choose an initial analyticit y radius ρ (0) suc h that ρ (0) ≥ Z + ∞ 0 k z ∂ z V ( t, z ) k L ∞ ( R + ) d t. (11.5) A priori , there is no reason for this integral to b e finite, so we will need to adapt our construction in order to ensure it. 12. Preparing a go o d-enough dissipation W e now turn to the first problem men tioned in Section 9: namely the fact that the b ound- ary lay er term V is not small enough at the final time T /ε . W e wish to c ho ose the source term h of (8.2) more wisely in order to ensure that V decays sufficien tly fast. As an added b enefit, this will make the integral in (11.5) finite. F or t ≥ T , h ( t ) = 0 so (8.2) is a free heat equation with n ull b oundary condition at z = 0. The decay rate of the free heat equation on the half-line is linked with the lo w frequencies of the “initial” data V ( T , · ). More precisely , it dep ends on the n umber of v anishing deriv atives of its F ourier transform at zero. These quan tities are linked to the z -momen ts R + ∞ 0 z k V ( T , z )d z , whic h are link ed with the t -moments of the source term h , R T 0 t k h ( t )d t . F or example, c ho osing h ∈ C ∞ ([0 , T ]; R ) such that R T 0 h ( t )d t = 0 guarantees that R + ∞ 0 z V ( T , z )d z = 0, which in turn improv es the deca y rate of the solution by a factor 1 /t for t → + ∞ . The k ey idea here is th us to choose a function h ∈ C ∞ ([0 , T ]; R ) whic h has a finite n um- b er of null time momen ts. This guarantees that the solution to (8.2) will decay sufficien tly fast (not only in L 2 ( R + ) but also for stronger functional spaces including p olynomial w eights in z and Sobolev norms). As a consequence, we obtain     V  T ε      L 2 ( R + ) = O ( ε 3 ) and Z + ∞ 0 k z ∂ z V ( t, z ) k L ∞ ( R + ) d t < + ∞ . (12.1) 13. Killing the initial data when it is outside The initial intuition, depicted in Figure 2 was to choose R T 0 h ( t )d t ≥ 3 L in order to flush the initial vorticit y ω ∗ outside of the physical domain Ω. Now that w e need to choose R T 0 h ( t )d t = 0, this intuition is not sufficien t an ymore. Ho wev er, we can use con trols to suppress the initial vorticit y while it is outside of the physical domain. F or example, is we c ho ose h suc h that R T / 3 0 h ( t )d t = 3 L , h = 0 on ( T / 3 , 2 T / 3) and R T 2 T / 3 h ( t )d t = − 3 L , w e hav e a reference flow which is globally of zero av erage, but for whic h there exists an intermediate time when the initial vorticit y ω ∗ is fully outside of the ph ysical domain. During ( T / 3 , 2 T / 3), we thus apply a con trol (in the form of a source term in (7.4), supp orted outside of Ω), whic h is designed to obtain u 1 (2 T / 3) = 0. Hence, when h b ecomes negativ e and “brings bac k” fluid particles in to Ω, it carries only a v anishing vorticit y . Heuristically , one sets u 1 ( t, x, y ) = β ( t ) u ∗  x − Z t 0 h ( t 0 )d t 0 , y  , (13.1) 8 NA VIER-STOKES CONTROLLABILITY where β ∈ C ∞ ( R + ; [0 , 1]) is suc h that β ( t ) = 1 for t ≤ T / 3 and β ( t ) = 0 for t ≥ 2 T / 3. This defines a solution of (7.4) with a non-zero right-hand side f 1 supp orted in [ T / 3 , 2 T / 3] × [2 L, 5 L ] × [ − 1 , 1]: f 1 ( t, x, y ) := β 0 ( t ) u ∗  x − Z t 0 h ( t 0 )d t 0 , y  . (13.2) T echnically , one should write a formula like (13.1) on the stream function in order to preserv e the divergence-free condition. 14. Dealing with the non-linearity with Chemin’s metho d An imp ortant drawbac k of the c hange of unkno wn (11.1) is that it destroys the nice structure of the nonlinear term ε ( r ε · ∇ ) r ε in equation (9.2). Usually , this term disapp ears during the standard L 2 energy estimate obtained by multiplying equation (9.2) by r ε and using the divergence-free condition. This simplification do es not happ en anymore after our change of unknown and w e m ust estimate this term. Using an idea in tro duced by Chemin in [1], w e no w see ρ as the unknown solution to the highly nonlinear ODE ρ 0 ( t ) = −k z ∂ z V ( t, z ) k L ∞ ( R + ) − ε k∇ r ε ρ ( t ) k ˙ B 0 2 , 1 , (14.1) where ˙ B 0 2 , 1 is a homogeneous Beso v space asso ciated with frequency truncations in the tangen tial direction and is designed to hav e the critical Sob olev embedding in tw o dimen- sions ( r ε ρ , ∇ r ε ρ ) ∈ ˙ B 0 2 , 1 ⇒ r ε ρ ∈ L ∞ . Exploiting the divergence-free condition on r ε ρ and this new definition of ρ 0 allo ws to control the nonlinear term. Ho wev er, since (14.1) is a nonlinear ODE, it is not clear a priori that its solution sta ys w ell defined (and p ositive) up to the final time T /ε (one could ha ve ρ → −∞ in finite time and we need to ensure ρ > 0 to stay within the analytic setting). Hence, we m ust p erform a parallel estimate for ρ in the same time as w e are estimating r ε ρ . Here, we use the viscous term − ε ∆ r ε of equation (9.2). Indeed, by Cauch y-Sch warz, the total decay of ρ can b e b ounded as ε Z T /ε 0 k∇ r ε ρ k ˙ B 0 2 , 1 ≤ √ T ε Z T /ε 0 k∇ r ε ρ k 2 ˙ B 0 2 , 1 ! 1 2 , (14.2) and the righ t-hand side is precisely the t yp e of quantit y for whic h w e obtain b ounds thanks to the viscous term − ε ∆ r ε when we p erform ˙ B 0 2 , 1 energy estimates on (9.2). W orking with ˙ B 0 2 , 1 (rather than L 2 ) is necessary in our context to b enefit from the em- b edding men tioned abov e. The nonlinear term r ∇ r is then estimated using paradifferen tial calculus techniques (including Bony’s parapro ducts), notably inspired by [2, 17]. 15. Sp otting the phantoms A drawbac k of the analytic setting considered ab o v e is the use of a phantom force (in the sense of a source term supp orted everywhere, arbitrarily small in an arbitrarily strong Sob olev space) for t wo different purp oses, whic h w e rev eal here. Of course, our strategy also requires large source terms (the controls) which are exclusiv ely supp orted outside of the physical domain Ω. Analytic regularization of the initial data. First, we need the initial data u ∗ to b e analytic. Since Theorem 4.1 is stated with an L 2 initial data, we need a strategy to regularize it. It is well kno wn that the Na vier-Stokes equation exhibits a strong smo othing 9 J.-M. Cor on, F. Marba ch, F. Sueur, & P. Zhang effect (thanks to the dissipation term) and that the solution instan tly b ecomes analytic. Ho wev er, the analytic radius at time t > 0 is only known to grow lik e √ t . Since we seek a small-time control result, the natural smo othing only yields a small analyticit y radius. Ho wev er, the total loss of analytic radius in our setting is linked with the quantit y Z + ∞ 0 k z ∂ z V ( t, z ) k L ∞ ( R + ) d t. (15.1) In turn, this quantit y dep ends on L and T through the choice of the base flow h . It can b e c heck ed that since we require R T / 3 0 h ≥ 3 L , the quan tit y (15.1) is b ounded b elow. Hence, as a first step of our result, we use an external source term supp orted everywhere to trim off the high tangential frequencies of the initial data and make it analytic with a sufficient radius (say twice the v alue of (15.1)). Since our metho d only needs to know that the analytic radius is large enough (and not that the asso ciated analytic norm of the initial data is small), this clipping pro cess can b e done with a small source term ev en in a strong Sob olev space, ensuring (4.1). Almost compactly supp orted extension. Second, lo oking at (13.2) defining the ex- ternal force used to driv e u 1 to zero, one sees that its size within Ω is link ed to the v alues of the extension u ∗ in [ − 3 L, − 2 L ] × [ − 1 , 1]. Our initial idea was to choose u ∗ compactly sup- p orted, sa y in [ − L, 2 L ] × [ − 1 , 1]. Of course, since we need u ∗ to b e analytic in the tangen tial direction, it cannot simultaneously ha ve a compact supp ort in x . The most we can require is that the Sob olev norm of the analytic extension u ∗ is small in [ − 3 L, − 2 L ] × [ − 1 , 1]. Then, from (13.2), we see that f 1 can b e split as a control part (large, but supp orted outside of Ω) and a phantom part (small, but supported inside Ω). In fact, our detailed construction also pro v es that we can lo calize the supp ort of this second phan tom force in the vertical direction so that it do es not touch the horizon tal b oundaries Γ ± . More precisely , for every T > 0 and u ∗ ∈ L 2 (Ω), w e prov e that there exists δ > 0 such that, for an y k ∈ N and η > 0, we can maintain the result of Theorem 4.1 while ensuring that supp f 1 | Ω ⊂ [0 , L ] × [ − 1 + δ, 1 − δ ]. This highlights the fact that the main role of this second phantom force is to allow us to work in an analytic setting (but not to tak e care directly of the b oundary la yer). Bibliograph y [1] Jean-Yves Chemin. Le syst` eme de Navier-Stok es incompressible soixan te dix ans apr` es Jean Leray . In A ctes des Journ´ ees Math´ ematiques ` a la M´ emoir e de Jean Ler ay , volume 9 of S´ emin. 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[16] Marco Sammartino and Russel Caflisch. Zero viscosity limit for analytic solutions of the Navier- Stok es equation on a half-space. I I. Construction of the Navier-Stok es solution. Comm. Math. Phys. , 192(2):463–491, 1998. [17] Ping Zhang and Zhifei Zhang. Long time w ell-p osedness of Prandtl system with small and analytic initial data. J. F unct. Anal. , 270(7):2591–2615, 2016. Jean-Michel Coron Lab oratoire Jacques-Louis Lions, Sorb onne Univ ersit´ e, F rance coron@ann.jussieu.fr Fr ´ ed ´ eric Marbach Univ Rennes, CNRS, F rance frederic.marbac h@ens-rennes.fr Franck Sueur Institut de Math´ ematiques de Bordeaux, Univ ersit´ e de Bordeaux, F rance franc k.sueur@math.u-b ordeaux.fr Ping Zhang Academ y of Mathematics & Systems Science and Hua Lo o-Keng Key Lab oratory of Mathematic, The Chinese Academy of Sciences, China zp@amss.ac.cn 11